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Graduate School University of Goteborg GMO0751 Advanced Microeconomic Theory Make-Up Exam (Autumn 2022) Instructors: C.A. Nadeau, K. Bolin Date: August 23, 2023 This exam will account for 100% of your final grade in the course and consists of 5 questions worth 100 points total. You are allowed to use the following material: a hand calculator and a dictionary. Please show all your work in order to receive full credit for each question; partial credit will be given if only minor mathematical errors are made in your calculations. Think carefully about each question before answering it — the quality of your answer is far more important than its length. Good luck! (1) (2) (3) A consumer purchases n commodities where the price of commodity i is pi fori=1, 2, .., n. Letthe consumer’s compensated and ordinary demand functions for commodity i be given, respectively, as: Hf(pl, P2, ., Pn, U) D](pll P2, ..., Pn, Y) where u denotes a target level of utility and y denotes income. (a) Is it possible to have 8H//dpi > 0? Carefully explain using both the lecture and textbook material. (b) Is it possible to have dD/dpi > 0? Carefully explain using both the lecture and textbook material. (c) Is it possible to have both OH'/dp; > 0 and dHi/dpi< 0? Carefully explain. (d) Is it possible to have both 3D'//dp; >0 and 0Di/dpi < 0? Carefully explain. (20 points total) The output level (q) for a competitive firm is determined by log(q) = Zailog(z) where z denotes its usage of input i and a; > 0 denotes a parameter wherei=1, 2, ..., m. (a) Derive the long-run average and marginal cost functions. Will marginal cost rise with output? (b) Derive the short-run marginal cost function if only k of the minputs are variable. (c) Derive the short-run elasticity of supply and then show what happens to this elasticity if k is reduced. (20 points total) Anna’s preferences can be represented by the following Cobb-Douglas utility function: U(C, L) = C¥5LY5 where C denotes consumption and L denotes leisure. Assume the price of consumptionis 1, Anna can work in the labour market at a 50 SEK hourly wage rate and that Anna divides 40 hours per week total to labour and leisure. (a) How many hours of leisure will she choose? (b) How would she have chosen if her utility function were U(C, L) = cy2L Y2 (20 points total)
(4) (5) An exchange economy is comprised of two individuals, A and B, and two goods, x1 and x2. The individuals’ preferences are represented by the following utility functions: UAxA, x2%) = xifx® and UB(xi%, x2B) = xa® + In(x2°) where xi' and x2 are individual 's consumption of good 1 and good 2, respectively. The initial endowments of the two goods are, for individual A: wi* =10 and w2* = 20, and for individual B: w1® =20 and w.® = 26. (a) Calculate the price ratio that gives a Walras equilibrium. (b) How would this equilibrium price ratio change if individual A’s initial endowment of good 1 increases by 10 units (i.e. what is the effect on p/p2 that gives a Walras equilibrium if wit = 20)? (20 points total) Knut divides his life into two different periods. His preferences for consumption in period one and period two can be represented by the following utility function: U(cy, €2) = In(ca) + [1/(1+p)lin{cz) , p>0 where c1 and c2 denote consumption in the first and second period, respectively, and p is a subjective discount factor. Knut's income is the same in both periods (i.e. y1 = y2 = y) and he has access to perfect financial markets. The rate of interestis r > 0. Derive Wsumption in the two periods as functions of r, p and y. Is Knut saving, borrowing or neither if p =r (note: your answer must be motivated and assume thaty = 1)? (20 points total)